We simulate 1000 years of the earthquake cycle along the San Andreas Fault System by convolving best estimates of interseismic and coseismic slip with the Green's function for a point dislocation in an elastic plate overlying a viscoelastic half-space. Interseismic slip rate is based on long-term geological estimates while fault locking depths are derived from horizontal GPS measurements. Coseismic and postseismic deformation is modeled using 70 earthquake ruptures, compiled from both historical data and paleoseismic data. This time-dependent velocity model is compared with 290 present-day geodetic velocity vectors to place bounds on elastic plate thickness and viscosity of the underlying substrate. Best fit models (RMS residual of 2.46 mm/yr) require an elastic plate thickness greater than 60 km and a substrate viscosity between 2 ¿ 1018 and 5 ¿ 1019 Pa s. These results highlight the need for vertical velocity measurements developed over long time spans (>20 years). Our numerical models are also used to investigate the 1000-year evolution of Coulomb stress. Stress is largely independent of assumed rheology, but is very sensitive to the slip history on each fault segment. As expected, present-day Coulomb stress is high along the entire southern San Andreas because there have been no major earthquakes over the past 150--300 years. Animations S1 and S2 of the time evolution of vector displacement and Coulomb stress are available as auxiliary material.